The level of detail available in reservoir description often exceeds the computational capability of existing reservoir simulators. This resolution gap is usually tackled by upscaling the fine-scale description to sizes that can be treated by a full-featured simulator. In upscaling, the original model is coarsened using a computationally inexpensive process. In flow-based methods, the process is based on single-phase flow. A simulation study is then performed using the coarsened model. Upscaling methods such as these have proven to be quite successful. However, it is not possible to have a prior estimate of the errors that are present when complex flow processes are investigated using coarse models constructed via these simplified settings.
Various fundamentally different multi-scale approaches for flow in porous media have been proposed to accommodate the fine-scale description directly. As opposed to upscaling, the multi-scale approach targets the full problem with the original resolution. The upscaling methodology is typically based on resolving the length and time-scales of interest by maximizing local operations. Arbogast et al. (T. Arbogast, Numerical subgrid upscaling of two phase flow in porous media, Technical report, Texas Institute for Computational and Applied Mathematics, The University of Texas at Austin, 1999, and T. Arbogast and S. L. Bryant, Numerical subgrid upscaling for waterflood simulations, SPE 66375, 2001) presented a mixed finite-element method where fine-scale effects are localized by a boundary condition assumption at the coarse element boundaries. Then the small-scale influence is coupled with the coarse-scale effects by numerical Greens functions. Hou and Wu (T. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comp. Phys., 134:169-189, 1997) employed a finite-element approach and constructed specific basis functions which capture the small scales. Again, localization is achieved by boundary condition assumptions for the coarse elements. To reduce the effects of these boundary conditions, an oversampling technique can be applied. Chen and Hou (Z. Chen and T. Y. Hou, A mixed finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comput., June 2002) utilized these ideas in combination with a mixed finite-element approach. Another approach by Beckie et al. (R. Beckie, A. A. Aldama, and E. F. Wood, Modeling the large-scale dynamics of saturated groundwater flow using spatial filtering, Water Resources Research, 32:1269-1280, 1996) is based on large eddy simulation (LES) techniques which are commonly used for turbulence modeling.
Lee et al. (S. H. Lee, L. J. Durlofsky, M. F. Lough, and W. H. Chen, Finite difference simulation of geologically complex reservoirs with tensor permeabilities, SPERE&E, pages 567-574, 1998) developed a flux-continuous finite-difference (FCFD) scheme for 2D models. Lee et al. further developed a method to address 3D models (S. H. Lee, H. Tchelepi, P. Jenny and L. Dechant, Implementation of a flux continuous finite-difference method for stratigraphic, hexahedron grids, SPE Journal, September, pages 269-277, 2002). Jenny et al. (P. Jenny, C. Wolfsteiner, S. H. Lee and L. J. Durlofsky, Modeling flow in geometrically complex reservoirs using hexahedral multi-block grids, SPE Journal, June, pages 149-157, 2002) later implemented this scheme in a multi-block simulator.
In light of the above modeling efforts, there is a need for a simulation method which more efficiently captures the effects of small scales on a coarse grid. Ideally, the method would be conservative and also treat tensor permeabilities correctly. Further, preferably the reconstructed fine-scale solution would satisfy the proper mass balance on the fine-scale. The present invention provides such a simulation method.